Brehm #3 Flashcards
Parameter risk
risk of assuming incorrect distributions or parameters for those distributions
Components of parameter risk (3)
- estimation risk
- projection risk
- model risk
Estimation risk
uncertainty arising from having only a sample of possible data to estimate parameters
Largest source of parameter risk
projection risk
Coefficient of variation of total claims and what it measures
CV ( S ) = [ ( Var ( N ) / E [ N ] + CV ( X )^2 ) / E [ N ] ] ^ .5
*measure of projection risk
larger CV = more risk
Relationship b/w size and risk for CV
more risk for smaller companies
however, as projection risk increases, rises more for large companies because small companies are already volatile
Components of projection uncertainty (2)
- uncertainty associated with historical data
2. uncertainty in fitted trend line
Relationship b/w claim severity trend and general inflation
claim severity trend > general inflation
> > excess is called social inflation or superimposed inflation
Relationship b/w uncertainty and time when analyzing trend
prediction intervals widen over time (b/c of uncertainty in the trend rate)
Method to assess estimation risk
use the covariance matrix from MLE procedure and assume the parameters come from a joint lognormal distribution
What is a copula?
combination of individual marginal distributions into a multivariate distribution to force correlation in specific areas
F ( x, y ) = Pr ( X < x and Y < y ) = C ( u, v )
or = C ( F ( x ), F ( y ) )
where u, v are percentiles of X and Y
Simulation process using copulas (3 steps)
- random draws, u and p, where p is a draw from the conditional distribution so p = C-sub 1 ( u, v )
- invert C-sub 1 by solving for v
- invert marginal X and Y distributions by setting F ( x ) = u and F ( y ) = v and solving for x and y
Which copulas (3) are invertable and why does this matter?
- Frank
- Normal
- HRT
** if copulas are invertable, they can be simulated
Common copula ranks in terms of right tail correlation (least to most)
Frank < Normal < Gumbel < HRT
Purpose of tail concentration functions / L-R graph
provides a quantification of tail strength
> > L ( z ) and R ( z ) are measures of correlation
L-R graph
L ( z ) and R ( z ) against z which exists on [0,1]
Left tail function - L ( z ) and lower tail dependence parameter
L ( z ) = C ( z, z ) / z
lower tail dependence parameter = limit of L ( z ) as z»_space; 0
Right tail function - R ( z ) and upper tail dependence parameter
R ( z ) = ( 1 - 2 * z + C ( z, z ) ) / ( 1 - z )
upper tail dependence parameter = limit of R ( z ) as z»_space; 1
Methods to model projection risk (2)
- Simple fixed trend model
- Auto correlated time series
Simple trend model understates projection risk especially for long tailed lines
Methods to compare tail behavior between 2 copulas (2)
- Left-right tail dependence (L-R graphs)
2. Plot joint density functions and compare density in the tails
Components of an internal risk model (IRM - 4)
- Startup
- Parameter development
- Implementation
- Integration and maintenance
Recommendations for IRM startup (4)
- Reporting relationship - fair and balanced leader more important than functional reporting line
- Resource commitment - establishing a new competency - best to transfer or hire full time employees
- Control of inputs and outputs
- Initial scope - prospective UW period and variation around plan
Recommendations for parameter development in an IRM (4)
- Modeling software - capabilities, scalability, and integration with other systems as well as user capabilities
- Parameter development - involves many functional areas (UW, claims, planning, and actuarial), heavily data driven but requires expert opinion - ensure clear ownership of final parameters
- Correlations - cannot set cross lines parameters in isolation, should have corporate level ownership
- Validation - test over an extended time period
Recommendations for implementation of an IRM (4)
- Priority setting - should come from sr management (more efficient)
- Communication - plan for regular communications to a broad audience
- Pilot testing - prepare company for magnitude of change
- Education - bring leadership to a base level of understanding
Recommendations for integration and maintenance for an IRM (3)
- Cycle - integrate into corporate calendar, especially into planning process
- Updating - major reviews not more than semi-annually
- Controls - maintain centralized control of inputs, outputs, and possibly application templates
